Molecular potential and elastic scattering of alpha particles by 28Si from 14 to 28 MeV

Nuclear Physics A504 (1989) 130-142 North-Holland, Amsterdam

M O L E C U L A R P O T E N T I A L A N D E L A S T I C S C A T T E R I N G OF

ALPHA PARTICLES BY 2SSi FROM 14 TO 28 MeV P. MANNG,~RD ~, M. BRENNER ~, M.M. ALAM2, I. REICHSTEIN 3 and F.B. MALIK 1'2 i Department of Physics, Abo Akademi, Porthansgatan 3, 20500 Abo, Finland Physics Department, Southern Illinois University, Carbondale, Illinois 62901, USA 3 School of Computer Science, Carleton University, Ottawa, Ontario K1S5B6, Canada Received 20 March 1989 (Revised 17 May 1989)

Abstract: The real part of the alpha-2SSi potential has been derived in the energy density formalism using the sudden approximation. The potential is found to be of the molecular type. The angular distributions at 14.47, 21.9, 24.0, 25.0, 26.0, 26.5, 27.0 and 28.0 MeV (lab) have been calculated using potentials, the real part of which are similar to the calculated one. The calculations using the molecular type of potentials can adequately account for such broad features as magnitudes, locations of maxima and minima and back-angle enhancement in angular distributions. In fact, in the energy region of 21.9 to 28.0 MeV (lab) the above observed features can be understood using an energy-independent real part. An analysis of the S-matrix using this potential does not reveal any obvious resonance, but implies that interference among partial waves determines the structure of the angular distributions. An alternative explanation for ALAS in 28Simight be that the potential between alpha particles and 28Si is of the molecular type, having a soft short-range repulsive core. E

NUCLEAR REACTION 2sSi(a, a), E = 14.47 MeV; measured o-(0). Energy density formalism, molecular potential, other data analyzed.

1. Introduction Since the early e x p e r i m e n t of Corelli et al. 1) discovering u n e x p e c t e d large backward e n h a n c e m e n t o f the elastic scattering of 18 MeV a l p h a particles by 160, such effects have b e e n observed for m a n y targets up to 48Ca. This effect, called a n o m a l o u s large angle scattering (ALAS) has also b e e n seen in the case of a l p h a scattering by 28Si [refs. 2-6)]. A recent systematic e x p e r i m e n t a l study u n d e r t a k e n with the .~bo A k a d e m i cyclotron confirms this effect in the energy range o f 12 to 18 MeV. Analyses o f the data have b e e n d o n e u s i n g a variety o f a p p r o a c h e s with limited success. Wiihr et al. 5) used a n optical potential with a H a u s e r - F e s h b a c k type of r e s o n a n c e term a d d e d i n c o h e r e n t l y to it to analyze their energy averaged data b e t w e e n 15.0 a n d 16.4MeV. Obst et al. 3) used a c o u p l e d - c h a n n e l s code a l o n g with angularm o m e n t u m - d e p e n d e n t a b s o r p t i o n to analyze the 25.2, 26.2 a n d 27.2 MeV data. Lega et al. 4) a d d e d an a n g u l a r - m o m e n t u m - d e p e n d e n t a b s o r p t i o n to Wiihr et al. 5) a p p r o a c h a t t e m p t i n g to fit the 21.9 MeV data. Jarczyk et al. 7) used a n optical m o d e l with a H a u s e r - F e s h b a c k term a d d e d to it to fit their data. All these attempts have 0375-9474/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

P. Manngdrd et aL / Molecular potential

131

had only limited success. The type of potential used by Michel et aL 8) in fitting the alphaM60 and alpha-4°Ca [refs. 21,22)] scattering data seems to be suitable at energies higher than those considered here. The purpose of this paper is to investigate whether or not a single theoretical approach can explain data in the energy range of 14 to 28 MeV. For this purpose we calculate the alpha-28Si potential using the energy density formalism 9-12) in the sudden approximation. Using this calculated potential as a guide we determine the best set of potentials that fit angular distributions at individual energies in the range of 14 to 28 MeV. We have selected data only at those energies where a fairly complete angular distribution is available. As noted by Budzanowaki et al. ~3), data either only at forward or at backward angles are insufficient to determine potentials. We, then, proceed to find a unique set of parameters for the real part of the potential (i.e., energy-independent real part) that explains the broad features such as the magnitude, ALAS, the approximate number of maxima and minima at about the observed angles in the entire energy range and compare this potential with the one calculated from the energy density formalism and those used for fitting the data at a particular energy. In sect. 2 we present a brief outline of the calculation of the potential in the energy density approach. Sect. 3 deals with the analysis of the data along with discussion. Sect. 4 summarizes the conclusions. 2. Calculation of the potential We present here the outline of the energy density formalism used in obtaining the alpha-nucleus interaction. The details are given in refs. 10,12). The energy E of a system of fermions, for a given density distribution p(r) is given by E

=

f e[p(r)] d3r, d

(1)

where 4p(,-)I

=

+

q~c(r)pp- 0.739 e 2 pp+ 4

where x is the neutron excess, M is the mass adjusted to reproduce the nuclear masses. the medium is v(p, x) and is obtained from using a realistic two-nucleon potential. The by 9)

- -

8M

r / (Vp )

2

,

(2)

of a nucleon and ~/is the free parameter The average potential of a nucleon in a Brueckner, Hartree-Fock calculation density dependence of v(p, x) is given

v( p, x) = bl(1 + alx2)p + b2(1 + a2x2)p 4/3 + b3(1 +

a 3 x 2 ) p 5/3 .

(3a)

P. Manngdrdet al. / Molecularpotential

132

The C o u l o m b potential, q0c, is related to the p r o t o n charge d i s t r i b u t i o n , pp, as follows

q0c = e

f pp(r')d3r ' Ir - r'[

(4)

The fourth term is the e x c h a n g e correction to P c [ref. 14)]. As n o t e d in refs. ]~.12) a single choice of p a r a m e t e r s ~7, a l , bl, a2, b2, a3 a n d b3 can r e p r o d u c e the observed n u c l e a r masses at least as well as those o b t a i n e d from a n y s t a n d a r d mass formula. For the alpha-28Si case, x, the n e u t r o n excess is zero for the projectile, the target a n d the composite system. H e n c e v(p) reduces to v ( p ) = bl p + b2 p4/3 _f_b3 p5/3. The p o t e n t i a l

V(R)

(3b)

b e t w e e n a l p h a a n d silicon at a distance R is given by

V(R) = E[p(r,

R)]-

E,[p](r,

R)]-

ErEp2(r,

R)],

(5)

where p, Pl a n d P2 are, respectively, density d i s t r i b u t i o n f u n c t i o n s o f the composite system, a l p h a particle at R --0o a n d silicon target at R = co. I n the s u d d e n a p p r o x i m a t i o n

p(r) =pl(r)+p2(r).

(6)

Thus, once p] a n d P2 are specified, p at a s e p a r a t i o n R can be o b t a i n e d by simply a d d i n g the a l p h a a n d silicon density distributions. The p a r a m e t e r r / i s chosen to r e p r o d u c e the observed masses of nuclei from mass n u m b e r A = 16 to 62 u s i n g r e a s o n a b l e f u n c t i o n a l forms to r e p r o d u c e the observed density distribution. I n general we used only t w o - p a r a m e t e r Fermi or G a u s s distribution f u n c t i o n s n o t e d as F a n d G, respectively in c o l u m n 2 o f table 1. Parameters o f these f u n c t i o n s are listed in refs. 15.16). TABLE 1

Calculated and observed masses for selected elements for r/= 8 Element

Density distribution

Observed masses (MeV)

Calculated masses (MeV)

12C ~60 28Si

E :G E :G J : F set I J: F set II J :F J:F J : F set I J : F set II

-92.16 - 127.62 -236.5

-91.99 - 127.68 -233.3 -234.3 -342.8 -439.9 -550.4 -552.5

4°Ca 5°Cr 62Ni

-342.1 -435.0 -545.3

J and E refer, respectively, to ref. 15) and ref. ~6). G and F refer, respectively to a two-parameter Gauss and Fermi density distribution function.

P. Manngdrd et al. / Molecular potential

133

For the alpha particle, there is no two parameter Gauss or Fermi distribution function listed in ref. 15) or ref. 16). Although a single Gauss function does not fit the alpha density distribution very well, Elton 16) lists as reasonable p(r) = 4(3,/rr) 3/2 exp ( - 7 r 2 ) ,

(7)

with y = 0.5. The binding energy and root mean square radius of the alpha particle using (7) are, respectively -17.8 MeV and 1.73 fm which are acceptably close to the observed values of -28.3 MeV and 1.69-1.7 fro. Alternatively, 3' could be determined by minimizing the total energy E given by (1) with respect to 3'. This yields 3' close to 0.45, a root mean squared radius of 1.826 fm and a binding energy of -20.01 MeV. Thus, the binding energy improves at the expense of the r.m.s, radius. There are also two alternative sets of two parameter Fermi distribution functions proposed for 2SSi. One yields a central density Po--0.182 and the other po = 0.169. We have selected the first one. | n fig. 1 we have plotted two theoretically calculated potentials for ~ = 8. Parameters of 2SSi a r e those of the set I of De Vries et al. 15). This yields Po = 0.182 and a binding energy of -236.5 MeV. The curves with squares and circles, respectively, correspond to 3' = 0.5 and 0.45 and calculated Po and binding energies of the alpha-particle are 0.254 and -17.8 MeV and 0.217 and -20.0 MeV, respectively. As expected the proper po for the alpha particle makes the potential more repulsive whereas, a density distribution yielding a better binding energy produces a slightly deeper minimum.

20 >

o

~

0

r a d i u s (fin) Fig. 1. Theoretically calculated real part of the potential for two sets of density distribution of the alpha particle and set I of ref, 15) for 2SSi. Squares and circles correspond respectively, to ~, = 0.50 and 0.45 in eq. (7). The solid line is the standard real potential defined by eq. (11).

134

P. Manngdrd et al. / Molecular potential

3. Analysis and discussion The discussion at the end of the last section indicates that the details of the alpha-silicon potential is somewhat sensitive to the alpha-particle density distribution functions. Nevertheless, the theory indicates that the potential is of the molecular type having a repulsive core at short distance with a height of about 15-20 MeV and half width of about 2.5 fm. This repulsion is followed by an attractive part with a minimum of about 8-10 MeV around a separation of 4 fm and is about 3.5 fm wide at the half of its depth. In addition, the shape of the core is close to a Gauss function. Using this as a guide, we attempt to fit the angular distributions at lab energies 14.47, 21.9, 24.0, 25.0, 26.0, 26.5, 27.0 and 28.0 MeV. Only at these energies are fairly complete data on angular distributions available. We left out the 15.016.4 MeV data of ref. 5) since the published figure represents energy averaged data, and the data of ref. 2) since they cover mainly the same energy region as that by ref. 6).] We do not attempt to determine the parameters from a least-squared fitting procedure, but by visual means, since the absolute magnitudes of the cross-section at a given energy could have some uncertainties due to energy calibration and target thickness 13). The real and imaginary parts of the potential are parametrized as follows: =

- -

with

+Vlexp

-

+Vc,

(8)

r zlz2e2 ( 2) v~(R) = |z,z2 e 2 ( R '

R > Rc.

(9)

Although the potential (8) has six parameters, one should recognize that a molecular type of potential is actually characterized by four parameters, the core height and its width, the location of the minimum and the width of the well. The imaginary part of the potential, W ( R ) , is taken to be of the gaussian type W(R)

= - Wo(E)

exp

-

.

(10)

This is motivated by noting that a microscopic calculation of the imaginary part of the 160-160 potential 17) indicates this to be of the volume type and energy dependent. The form (10) has also been used for the elastic scattering of 160 by 160 and 12C by 12C [ref. 18)]. In table 2 we list the parameters used at different energies to obtain the best fit. The 21.9 MeV data have been analyzed with two sets of parameters. One set of parameters reproduces the overall pattern better but if one puts emphasis on

P. ManngJrd et al. / Molecular potential

135

TABLE 2 Parameters of the potential defined by eqs. (8), (9) and (11) used in obtaining the best fit to the data at different energies. The last column gives the calculated reaction cross section crr at each energy. E (MeV)

Vo (MeV)

V1 (MeV)

Ro (fm)

Rl (fm)

a (fm)

Wo (MeV)

Rw (fm)

crr (mb)

14.47 21.9 21.9 24.0 25.0 26.0 26.5 27.0 28.0

16.5 22.0 30.0 24.0 26.0 26.0 26.0 28.0 26.0

34.0 42.0 50.0 42.0 42.0 42.0 42.0 42.0 42.0

5.70 5.50 5.40 5.50 5.35 5.35 5.35 5.37 5.35

2.40 2.90 3.10 2.80 2.80 2.80 2.80 2.78 2.90

0.60 0.55 0.42 0.34 0.34 0.34 0.34 0.35 0.36

4.5 12.0 12.0 14.0 14.0 15.0 15.5 17.0 17.0

3.90 4.00 3.90 4.00 4.00 3.90 3.80 3.85 4.10

893 1266 1171 1255 1240 1212 1175 1243 1367

reproducing the two minima around 100 ° the other set is the better one. Except for the nuclear radius parameter R0, the parameters of the potential needed to analyze the data at energies 21.9 MeV and above are very similar. The 14.47 MeV data taken with a very fine energy resolution ( - 5 0 keV) and a thin target (60 ixg/cm 2) require a potential which is a few MeV shallower and slightly wider. The parameters listed for the 14.47, 24.0 and 28.0 MeV data differ somewhat from those reported in ref. 19) primarily because R c , the Coulomb radius, is taken to be slightly different in the two cases. At Rc = 7.1 fm as chosen in ref. ~9), the nuclear part of the potential still has a magnitude of about 0.5 MeV but for Rc = 9.35 fm chosen here, the nuclear part of the potential is de facto zero. The data could have been fitted with about the same accuracy with slightly different set of parameters if Rc is taken to be 7.1 fm. However, in that case in the region r-~ R c , the interaction would not be purely the C o u l o m b one. The fit to the 14.47 MeV data is satisfactory. Similar quality fits can also be obtained using W = -4.5 MeV and Rw = 3.9 fm, or W = - 7 . 0 MeV and Rw = 3.5 fm. However, there is not much room in choosing other parameters to get the same quality of fit. The theoretical angular distributions for both sets of parameters reproduce the m a x i m a and minima at about the correct angles for the 21.9 MeV data. In addition, except for the data around 100 °, the calculated and observed magnitudes are in reasonable agreement. Lega et al. 4) had difficulties in getting a satisfactory fit to these data taken by them. The data at 24.0-28.0 MeV [refs. 6,7)] are characterized by the fact that all angular distributions, except for the 27.0 MeV data, exhibit eight minima and seven maxima (not counting the forwardmost and the backwardmost rise). The 27.0 MeV data have an additional m a x i m u m and minimum. The data can be fitted satisfactorily, particularly in terms of magnitude at all angles, using almost the same set of parameters, except for the depth of the absorption and also the 27.0 MeV data

136

P. Manngdrd et al. / Molecular potential

)

a

10 I;

\

20.0

.~....~

..

,v

x 1 0 ~ . l ' ~'i. :,-,

10.0

.F 0.0

-10.0 o°,

10 20.0:

10.0

,., •"Q

1o

•

6

,1o'

-

.

:

.ij

"

.,..

,.

0.0

"a -1o.~i

b C~ 20.0 0

10

10.0

0.0

d)

-10.0

X1

"

;

10 0-

o

..

~.

"

20.0

10.0.

0.0

,

.

.

.

.

.

.

t

. . . . . . . . .

i

.

.

.

.

.

.

i

. . . . .

10.0-

w . . . . . .

50

100

OCM (deg)

150

i . . . . . . . .

2

i

4

q . . . . . .

1 .......

6

| . . . . . . . . .

8

10

radius (fm)

Fig. 2. To the left the observed angular distribution marked by solid dots are compared with those obtained using potentials defined by parameters listed in table 2 (marked by solid lines) and the standard potential, eq. (11) (marked by dotted lines). (a) to (i) refer, respectively, to 14.47, 21.9, 21.9, 24.0, 25.0, 26.0, 26.5, 27.0 and 28.0 MeV a kinetic energy in lab, corresponding to 12.66, 19.16, 19.16, 21.0, 21.88, 22.75, 23.19, 23.63 and 24.50 MeV in c.m., respectively. The 14.47 MeV data are from the present work and the 21.9 MeV data are from ref. 4), while the rest are from refs. 6,7). TO the right of the figure the real part of the standard potential, eq. (11) (dotted line) is compared to those needed to obtain the best fit to the data using parameters listed in table 2 (solid line).

a~/d~ (~b/~) i

i

[

i

i

I~

i

J

i~

I

I

i

I ,o

o.-, e-

.'e.e.

•

.

.

O~

0

real potential (MeV)

~.

.

.

I

I

I~

i

I

i~,

i

138

P. Manngdrd et aL / Molecular potential

requires a slightly larger Ro. In fig. 2a-i, the observed angular distributions marked by solid dots are compared with the calculated ones, marked by solid lines at 14.47, 21.9, 24.0, 25.0, 26.0, 26.5, 27.0 and 28.0 MeV (lab). The corresponding centre-of-mass energies are 12.66, 19.16, 21.00, 21.88, 22.75, 23.19, 23.63 and 24.50MeV. The theoretical angular distributions are calculated using the parameters noted in table 2 for each energy. We also give the calculated reaction cross sections in table 2, although, to the best of our knowledge, no experimental data are available. Because of the similarity of parameters needed to fit the data from 21.9 to 28.0 MeV, it is reasonable to look for a unique set of parameters that might reproduce the broad features of the data in this energy range. We expect the degree of absorption, Wo, to depend on energy. Such a unique set of parameters is likely to facilitate, (a) the understanding of the behaviour of the S-matrix with energy for each partial wave and, (b) scaling of the appropriate parameters to deduce the potential between alpha particles and neighbouring nuclei. Such a scaling has been successful in predicting 0~-32'34S elastic-scattering cross sections 19). Since we have now chosen Rc, the Coulomb radius, to be different, we expect our set of parameters here to differ somewhat from the earlier one. The selected parameters are Vo = 26 MeV,

Ro = 5.35 f m ,

VI= 42 MeV,

R1=2.8 f m ,

a = 0.34 f m , Rc =9.35 fm.

(11)

The range, Rw, of the imaginary part is taken to be 4.0 fm at all energies. This set of parameters is very close to the one of ref. 19) except for Rc and a. The strength of the imaginary part of the potential, W o ( E ) , is taken to be 1, 9, 13.5, 14, 14.5, 15, 15.5 and 18 MeV at 14.47, 21.9, 24.0, 25.0, 26.0, 26.5, 27.0 and 28.0 MeV, respectively. The potential (11), henceforth called the standard potential, is plotted in both figs. 1 and 2 and marked by solid lines. In fig. 1 this potential is compared with the calculated ones from the energy density formalism and they are similar. In fig. 2 this standard potential is compared to the one needed to obtain the best fit at a particular energy and they are very close (in fact they are identical at 25.0, 26.0 and 26.5 MeV). In fig. 2a-i, the calculated angular distributions using the standard potential (11) with Wo listed above are plotted by dotted lines and compared to the data (solid dots) and the ones obtained from the potential using the parameters tabulated in table 2 (solid line). The angular distributions using both sets of potentials are similar, significant differences occurring only at 14.47 and at 27.0 MeV. The depth of the imaginary part of the standard potential can reasonably be parametrized in the energy range 21.9-28 MeV by a parabolic function Wo(E)= W o ( I + C , E + C 2 E 2)

21.9~
(12)

with Wo = 20 MeV, C1 = 0.126 and C2 = 0.005. The calculated angular distributions using the standard potential (11) with (12) are close to those plotted by dotted lines in fig. 2a-i. The use of the standard energy-independent real potential (11) with the smooth energy-dependent absorptive potential (12) allows us to investigate the

P. Manng~rd et aL / Molecular potential

139

behaviour of the S-matrix for each partial wave l, noted as St, in the energy range of 21.9 to 28 MeV. In fig. 3 we have plotted Re St/ISt I = cos 2At, where A~ is the real part of the phase shift for t h e / t h partial wave, as a function of energy in the range of 21.9 to 28.0 MeV for the standard potential (11) along with (12). We note that the behaviour of the partial waves for the parameters providing the best fit as a function of energy is similar. In this energy range only about 9 partial waves contribute significantly to the cross sections. It is interesting to note that, (i) there is strong interference among Re St for various partial waves. For example, Re S~/ISj I is negative around 24 MeV f o r / = 0 , 2, 3, 4, 5 and 6 but positive for higher partial waves. (ii) Except for 1 = 0, Re Sl/IStl behaves 1,0

0,0

•

0

o o

-1,0 1,0

•

~

i

•

•

0

,.'F 0°0

o

o~

o

-1,0 1,0

ii

~

•

•

•

•

0

•

!

~

•

o

0,0

-I,0

22

24

26

E,~ (lab) in M e V Fig. 3. Calculated Re St/]S~I = cos 2At, AI being the real part of the phase shift for a given 1, using the standard potential, eq. (11), along with the parametrized imaginary potential, eq. (12), are plotted as a function of energy in the interval 21.9-28.0 MeV (lab). Solid dots, open circles and solid squares represent, respectively, l = 0, 1 and 2 in the upper, 3, 4 and 5 in the middle and 6, 7 and 8 in the lower insert.

P. Manngdrd et al. / Molecular potential

140

1,0":.

•

0

0

D

D

•

t

•

0

0,0 ©

-1,0 1,0~

I

~

l

l

l

l

l

l

l

l

l

~

l

l

l

l

~

l

l

~

l

l

l

l

l

l

l O

l

l

~

O

O

o,5

angular momentum

l

Fig. 4. Calculated Re S,/ISt] = cos 2A, and the reflection coefficient IStl =exp (-2/z,) (A, and tzt are, respectively, the real and imaginary part of the complex phase shift) for 24.0 MeV incident energy (lab) are plotted for the potential used to obtain the best fit in the upper and lower inserts, respectively, as a function of angular momentum L b e h a v e s s m o o t h l y in the e n e r g y range o f 21.9 to 28.0 M e V ruling out a n y r e s o n a n c e for these /-values a n d (iii) Re St/lSt[ c h a n g e s from - 1 to +1 going t h r o u g h zero at a b o u t 26 M e V for l = 0. This might be the o n l y c a n d i d a t e for a r e s o n a n c e a m o n g these six p a r t i a l waves. H o w e v e r , in that case the w i d t h o f the r e s o n a n c e is a very b r o a d o n e o f several MeV. At h i g h e r energies m a n y m o r e p a r t i a l waves c o n t r i b u t e to the differential cross section. In fig. 4, we have p l o t t e d a s a m p l e values o f [Stl for 24 MeV. As n o t e d b y Brink a n d T a k i g a w a 20) the s o m e w h a t fluctuating b e h a v i o u r o f the p a r t i a l - w a v e a m p l i t u d e s m i g h t be d u e to i n t e r f e r e n c e o f waves reflected at the e x t e r n a l C o u l o m b b a r r i e r a n d the internal b a r r i e r in o u r case o r i g i n a t i n g from centrifugal a n d m o l e c u l a r p o t e n t i a l . The d i p a s s o c i a t e d with l = 7 a n d 10 m i g h t be i n d i c a t i v e o f s o m e r e s o n a n c e like b e h a v i o u r . There is no strong / - a b s o r p t i o n w i n d o w . The difference b e t w e e n the s t a n d a r d a n d the p o t e n t i a l u s e d to o b t a i n the best fit to the 14.47 M e V d a t a is significant. H e n c e , in fig. 5 we have p l o t t e d Re S~/]St[ a n d ]St] for the 14.47 M e V d a t a for the p o t e n t i a l u s e d for the best fit. The interference is very strong at this e n e r g y w h i c h is the cause o f s h a r p m i n i m a a n d m a x i m a in the cross section. The reflection coefficient as a f u n c t i o n o f 1 has m o r e structure c o m p a r e d to the 24 M e V case. H o w e v e r , there is a g a i n no p r o n o u n c e d w i n d o w a n d c o n t r i b u tions f r o m all partial waves u p to l = 10 are i m p o r t a n t .

P. Manng~rd et al. / Molecular potential

141

1,0-I

¢,q

o.0 0

-1,0 1,0~

!

D

Q

!

Q

•

0

0,

•

o,5

01~

i , , , , , t l l l l , l l l l l , , l l l l , l l l l ,

o

s

1o

I

I5

angular momentum l Fig. 5. Calculated Re S,/IS, I=cos2A, and the reflection coefficient ]Stl=exp (-2/zl) for 14.47 MeV incident energy (lab) are plotted for the potential used to obtain the best fit in the upper and lower inserts, respectively, as a function of angular momentum I. Analysis of the behaviour of $1 might shed some light on the cause of the failure by us and others 6,7) to fit these data using a Michel-type potential 8,2t.22). At the energies considered here, the contribution to cross sections by lower partial waves, even l = 0, is significant. For low partial waves, the character of the sum of the nuclear and centrifugal potential i.e., the total effective potential, depends to some extent on the interior of the nuclear potential, which controls the location of the minimum. For higher energies, used in refs. 8,21), major contributors to the cross section are higher partial waves and hence the nature of the nuclear potential near the surface only becomes the determining factor. It is, therefore, quite likely that a molecular type of potential used here might be applicable to alpha scattering by other nuclei at higher energy as long as that potential and the Michel-Vanderpoorten potential have similar shape in the surface region. We intend to pursue this investigation in the future.

4. Conclusion

This analysis of the elastic-scattering data indicates that a molecular type of potential provides an adequate description of the alpha scattering by 28Si in the energy range 14-28 MeV. The potential required is close to the one derived from

142

P. Manngdrd et al. / Molecular potential

the energy density formalism in the sudden approximation, particularly in the range 21-28 MeV. The potential required to fit the 14.47 MeV data is a few MeV shallower and wider, probably implying a slight adiabatic situation 1o). A potential containing an energy-independent real part is sufficient for an adequate description of the main features o f the observed cross section in the energy range 21.9-28 MeV. The cross sections are primarily determined by interference, and not resonances, among various partial waves. The cause of interference is the reflection at the external Coulomb barrier and the internal static barrier of the molecular potential and the centrifugal barrier. One of us (F.B.M.) is thankful to the Fulbright Commission and Stiftelsen f6r ~,bo Akademi for awards and to the members of Physics department of Abo Akademi for warm hospitality. The authors would like to thank Dr. M. Siemaszko from the Institute of Physics of the Silesian University, Katowice, Poland for making their experimental data on a + 28Si available to us.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)

J.C. Corelli, U. Bleuler and J. Tendam, Phys. Rev. 116 (1959) 1184 F.W. Bingham, Phys. Rev. 145 (1966) 901 A.W. Obst and K.W. Kemper, Phys. Rev. C5 (1972) 1705 J. Lega and P.C. Macq, Nucl. Phys. A218 (1974) 429 W. Wiihr, A. Hoffmann and G. Philipp, Z. Phys. 269 (1974) 365 L. Jarczyk, B. Maciuk, M. Siemaszko and W. Zipper, Acta Phys. Pol. B7 (1976) 53 L. Jarczyk, M. Siemaszko and W. Zipper, Acta Phys. Pol. B9 (1978) 167 F. Michel, J. Albinski, P. Belery, Th. Delbar, Ch. Gr6goire, B. Tasiaux and G. Reidemeister, Phys. Rev. 28 (1983) 1904 K.A. Brueckner, J.R. Buchler and M.M. Kelly, Phys. Rev. 173 (1968) 944 I. Reichstein and F.B. Malik, Phys. Lett. B37 (1972) 344 I. Reichstein and F.B. Malik, Ann. of Phys. 98 (1976) 322 I. Reichstein and F.B. Malik, Condensed matter theories, vol. 1 (1985) p. 291 A. Budzanowski, L. Jarczyk, L. Kamys and A. Kapuscik, Nucl. Phys. A265 (1976) 461 D.C. Peaslee, Phys. Rev. 95 (1959) 717 H. De Vries, C.W. De Jaeger and C. De Vries, At. Data Nucl. Data Tables 36 (1987) 495 L.R.B. Elton, Nuclear sizes, (Oxford Univ. Press, 1961) D.R. Alexander and F.B. Malik, Phys. Lett. 1342 (1972) 412 Q. Haider and F.B. Malik, J. of Phys. G7 (1981) 1661 P. Manng~rd, M. Brenner, I. Reichstein and F.B. Malik, Proc. 5th Int. Conf. on nuclear reaction mechanisms, ed. E. Gadioli (University of Milan Press, 1988) p. 385 D.M. Brink and N. Takigawa, Nucl. Phys. A279 (1977) 159 F. Michel and R. Vanderpoorten, Phys. Lett. B82 (1979) 183 F. Michel, Phys. Rev. C13 (1976) 1446